Abstract
We study the relation between class mathcal {S} theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual Omega -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding tau -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to W_N free fermion correlators on the torus.
Highlights
N mi−1 of such theories, namely the SU (2) N = 2∗ gauge theory, was shown to be related to the elliptic form of the Painlevé VI equation [8]
In order to understand the correspondence between isomonodromy deformations and four-dimensional N = 2 supersymmetric gauge theories a central object is the Hitchin system [9], in terms of which it is possible to formulate Seiberg–Witten theory, describing the Coulomb branch of the theory [10]
IR of the Coulomb branch was asked since the early days of Seiberg–Witten theory, and the answer to this question was found to be that one has to split the times of the integrable system into “slow” and “fast” times, effectively deautonomizing the system in a consistent way: this corresponds, in the language of integrable system, to the so-called Whitham deformations [16,17]
Summary
In order to understand the correspondence between isomonodromy deformations and four-dimensional N = 2 supersymmetric gauge theories a central object is the Hitchin system [9], in terms of which it is possible to formulate Seiberg–Witten theory, describing the Coulomb branch of the theory [10] The appearance of such an object is best understood within the context of class S theories [11,12,13,14,15]: one obtains theories in this class by compactifying the AN−1 six-dimensional (2, 0) superconformal field theory on a Riemann surface g,n of genus g with n punctures, with punctures carrying additional information given by singular boundary conditions for the fields. The question of how this picture gets modified when one tries to follow the physics from the deep
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